# UNDERSTANDING LINEAR ALGEBRAIC EQUATIONS VIA SUPER CALCULATOR REPRESENTATIONS

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2 to an equation under cross representation transformations, was emphasised during the learning experience. A consideration of the best way to approach this led to the graphic calculator, since all three dynamically related representations arise naturally in that context (Kaput, 199). In addition, graphic calculators are more accessible to students than computers are in many schools (Kissane, 1995; Thomas, 1996) and this is a key advantage. However some have been sceptical of the value of the technology in secondary mathematics learning and continued research is needed in order to provide convincing evidence of how graphic calculators can be valuably employed in the mathematics curriculum. There is already a growing body of such research (e.g. Ruthven, 1990; Penglase & Arnold, 1996; Graham & Thomas, 1997) supporting their use in mathematics learning, but it is still restricted in terms of content area and use of the calculators facilities. Method The research described here forms the Korean part of a study of students in New Zealand and Korea (a high performing country on the TIMSS results), whose aim is to investigate the use of super/graphic calculators to improve students conceptual understanding of linear algebraic equations. This research was carried out during the period 5th 16th July, Subjects: The study involved one class of 35 Form students (aged 13 1 years) from a middle school in Seoul, Korea, who are not currently allowed to use any calculator in their classes, or for assessment, such as examinations. Thus while 7 of the subjects had used a calculator (but never in their standard mathematics lessons) none had ever used a graphic or a super calculator. The students had previously covered simplification of algebraic expressions and solving linear equations during the school year. This enabled us to use them as a stand alone single subject group to see what, if anything, they could gain by additional exposure to and linking with the alternative methods of approaching equations that the super calculator permits. Instruments: A module of work using the TI 9 super calculator was prepared. This contained a description of the basic facilities of the TI-9 and then showed how, using a Press, See, and Explanation format, linear equations can be solved in three different ways: algebraically, graphically, and numerically from a table of values. Two algebraic methods were given, using the TI 9 to solve the equation directly and also using a standard balancing algorithm. An illustrative section from the module showing the four methods for the equation x 5 = 3x 9 are given in Figure 1 (note the section is incomplete and formatting has been changed). The fact that the solution is the same in each case was emphasised. Two parallel tests, divided into sections A and B and comprising different numerical values, were constructed as pre and post tests. Section A of these tests comprised standard textbooks questions such as: 5x 8 = 3x + ; m = 8 3m; and 6 8n = 3 + n. In contrast, section B addressed the students conceptual thinking in solving equations, both within and across different representations. The concept of equivalent equations is also important and we wanted to know whether the students were able to conserve equation under addition of constant or variable quantities as used in method 1b), and could recognise equivalent ones without having to find solutions.

3 x 5 = 3x 9 is solved by an algebraic method. The b x tells the calculator to solve with respect to x. x = is the value which makes both sides equal in value. Figure 1: A section of the module showing the layout and calculator screens To find the value of x, we need to simplify the given expression step by step: If we add 5 to both sides, the expression is simplified to x=3x-. If we subtract 3x, the expression is simplified to x =. If we divide by 1, finally we get x= Here each side of the equation is defined as a function, using y1(x) and y(x): y1(x)= x 5 y(x)= 3x 9 Looking at the two graphs, we can see that they intersect at one point. 1 st curve means y1(x), nd curve means y(x). The lower and upper bound means the interval in which the intersection point is found. So the two graphs intersect at the point (, 3). i.e. x = The point of intersection can be found using a table. Enter y1 and y as in method. When we look at the point x=, we can see the values of the two functions are the same, and equal to 3. Within the symbolic representation we asked them questions (B1) such as: Do the following pairs of two equations have the same solution? Give reasons for your answer. a) 5x 1 = 3x + b) 3x = x 3 5x = 3x x = x 3 We note that in both examples above the second equation may be seen by someone with the concept of equivalence of equations as a legitimate transformation of the first (by adding 8 or x to both sides) conserving the solution, although this may not be a productive transformation in terms of actually obtaining that solution. In addition the tests required the students to maintain the concept of solving an equation across representations, asking them to solve a symbolically presented equation in a graphical and a tabular domain (question B7) and to convert a graphical picture into a symbolic representation, as in Figure.

4 B6. Write a single equation in x which can be represented by the following diagram: y x Figure : A question assessing understanding of transformation from a graphical to a symbolic representation Students were also given a list of 6 symbolic, 3 graphical and 3 tabular representations and asked Which of the following are different ways of representing the same equations? (see Figure 3). This tested their ability to conceptualise functional equality across representations. B. Which of the following are different ways of representing the same equations? Match the letters which correspond and write them in the boxes below. x + = x x = 3 x +3 = x A B C 5 x = x x = x x = D E F G H I x y x y x y x y x y J K L - Answer. correspond correspond correspond correspond Figure 3. A question testing equivalent representations of equation forms As well as the test each student was given a questionnaire covering their experience with the calculators (serving to triangulate the test results) and their understanding of the concepts associated with linear equations, and an attitude scale (using a 5 point Likert format) on their feelings towards calculators. Procedure: The module (written in English and translated into Korean the English version is shown in this paper) was initially given to the class teacher, who familiarised herself with the content. The first named researcher met twice with her to answer her questions and to make sure that she was comfortable with the

6 equation, but on the post test he has made the link to a solution in each case, and understands why it is the solution. Table 1: A comparison of pre and post test section A and B mean scores Question Number (Max score) Pre test mean Post test mean A1 (5) n.s. A () n.s. B1 (18) n.s. B () <.0005 B3 () n.s. B (1) n.s. B5 () n.s. B6 () <.05 B7 (6) <.005 We note that for the graph question he also solves the equation symbolically, but does not do so for the table. Pre Test Solution Post Test Solution t p [x equals 1, because 1.00 is indicated on the x-axis] No Response [x = 1, because the values of y as at the point x = 1 are the same from the two tables] Figure : The B7 work of student L showing conceptual links between representations The aspect of conceptual understanding, seeing the relationship between equations and their solutions across the representations, was further investigated in the students questionnaire, where they were asked the following question: Is there a relationship between A, B, C in following diagrams? If so, then what is it? x + = x y x A B C This was very similar to question B, where the pre test facility was 11.5%, and the post-test 19%. In the questionnaire 17 (8%) of the students answered that the value of x is common, showing that they may have conserved solution of equation across the representations. It could be argued that they had merely calculated that in each case the solution was, without appreciating that the functions were the same. However, 9 (5%) of the students also answered that the expression A can be shown by the graph B and the table C, demonstrating that these had made the link. Further supporting this, 11 (31%) of the students mentioned later in their questionnaires, as x y x y

8 Conclusion Our contention is that important conceptual links between the symbolic, graphical and tabular representations of functions can easily be lost if algebra is approached in a purely procedural manner. The value of graphic and super calculators is that they may be used to assist teachers to make these links explicit, provided teachers are pedagogically alert to the deeper, underlying conceptual relationships, and preservation of the conceptual structure of the mathematics is central to their schemas. For example, one may approach the graphical solution of linear equations by providing the surface, procedural method of drawing the graph of each function and reading off the x value of the point of intersection without explicitly tackling the deeper, functional relationships (see Greer & Harel, 1998 for other examples). The concept of linear function passes across four different representations in this method: algebraic, tabular, ordered pairs and graphical. To build rich relational schemas which contain internal representations of the external ones, students should experience the links and the sub concepts of one to one, independent and dependent variable, etc. in each representation. In addition, the fact that two one to one functions coincide for a single value of the independent variable, and the conservation of these values across any representation is important. In this research study we have begun the process of testing the value of super calculators such as the TI 9 in promoting the construction of relationships like these. This small scale, uncontrolled study, provides some evidence of the value of the approach, with the students able to add conceptual schematic links to the knowledge they had built by studying equations purely symbolically. Whether the calculator is a significant factor, or the results could be duplicated without using them is of considerable interest, and we would welcome a study which sought to determine this. References Chinnappan, M. & Thomas, M. O. J. (1999). Structured knowledge and conceptual modelling of functions by an experienced teacher, Proceedings of the nd Mathematics Education Research Group of Australasia Conference, Adelaide, Graham, A. T. & Thomas, M. O. J. (1997). Tapping into Algebraic Variables Through the Graphic Calculator, Proceedings of the 1st Conference of the International Group for the Psychology of Mathematics Education, Lahti, Finland, 3, Greer, B. & Harel, G. (1998). The Role of Isomorphisms in Mathematical Cognition, Journal of Mathematical Behavior, 17(1), 5. Kaput, J. (1987). Representation systems and mathematics. In C. Janvier (Ed.) Problems of Representation in the Teaching and Learning of Mathematics (pp. 19 6). Lawrence Erlbaum: New Jersey. Kaput, J. (199). Technology and Mathematics Education, NCTM Yearbook on Mathematics Education, Kissane, B. (1995). The importance of being accessible: The graphics calculator in mathematics education, The First Asian Conference on Technology in Mathematics, Singapore: The Association of Mathematics Educators, Penglase, M. & Arnold, S. (1996). The Graphic Calculator in Mathematics Education: A critical review of recent research, Mathematics Education Research Journal, 8(1), Ruthven, K. (1990). The Influence of Graphic Calculator Use on Translation from Graphic to Symbolic Forms. Educational Studies in Mathematics, 1, Skemp, R. R. (1979). Intelligence, Learning and Action A Foundation For Theory and Practice in Education, Chichester, UK: Wiley. Tall, D. O., Thomas, M. O. J., Davis, G., Gray, E. & Simpson, A. (1999). What is the Object of the Encapsulation of a Process? Journal of Mathematical Behavior, 18(), Thomas, M. O. J. (1996). Computers in the Mathematics Classroom: A survey, Proceedings of the 19th Mathematics Education Research Group of Australasia Conference, Melbourne, Australia,

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